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Properties

Label 2-3456-9.4-c1-0-34
Degree $2$
Conductor $3456$
Sign $-0.766 + 0.642i$
Analytic cond. $27.5962$
Root an. cond. $5.25321$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)5-s + (−1 − 1.73i)7-s + (2.5 + 4.33i)11-s + (−2 + 3.46i)13-s − 17-s − 5·19-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + (−3 − 5.19i)29-s + 3.99·35-s − 10·37-s + (−1.5 + 2.59i)41-s + (4.5 + 7.79i)43-s + (−4 − 6.92i)47-s + (1.50 − 2.59i)49-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (−0.377 − 0.654i)7-s + (0.753 + 1.30i)11-s + (−0.554 + 0.960i)13-s − 0.242·17-s − 1.14·19-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + (−0.557 − 0.964i)29-s + 0.676·35-s − 1.64·37-s + (−0.234 + 0.405i)41-s + (0.686 + 1.18i)43-s + (−0.583 − 1.01i)47-s + (0.214 − 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.5 - 7.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.257867809038482960450034760515, −7.27288845220825905730133330062, −6.82607837403655934154670228704, −6.50833163270648095725746696478, −5.10263849895104316158316848147, −4.20029605006723500991586844923, −3.83654964063556896742550011938, −2.58457084481365404086027358371, −1.73478539784954574970057285916, 0, 1.16022226607341470172710149870, 2.51522742879910151141871075036, 3.43980589976707906914659552306, 4.16938209427224475275731329436, 5.29335547886563723879482722787, 5.68564364598820840199351247046, 6.63873940217139958063592763453, 7.42097593678275114819297366576, 8.452576685040873648601349959631

Graph of the $Z$-function along the critical line